I wouldn't worry too much about feeling uncoordinated, it's very normal as a beginner, everything will click into place soon enough and riding school horses are well used to it and won't come to any harm from the odd incorrect aid or unbalanced moment (they're trained to grind to a halt at any such thing which can be frustrating when you want to get going faster!). As others have said lunge lessons would be great and also yoga/pilates to build your core and general cardio fitness from walking/running/cycling/swimming is good too.
In terms of building a relationship with your horse, it's tricky as horses are not like dogs, they're not intrinsically motivated by human contact, words or praise so you need to learn how they tick and to speak their language more than human language, as it were. They tend to respond best to calm, consistent handling in a strong routine (so the same things always happen in the same order), and body language is very important, it's how they communicate with one another and they will respond strongly to it in people too.
The best thing you can do is spend as much time around horses as possible and observe what frightens them (most things lol), what makes them relax, what annoys them, how they interact with other horses and different people and so on. It doesn't have to be all riding time, being around them on the ground is just as good if not better. Appreciate this may be hard within the confines of a busy riding school but if you can ask to be shown how to get your horse ready before a lesson and look after him afterwards, so getting him out of the stable, leading, tie-ing up, grooming, tacking up, washing down and cooling off afterwards and so on would all be really useful skills for you - maybe if you can book a lesson at a quieter time of day your teacher or one of the yard staff would be able to show you how and then you can start doing this before and after your lessons. It would be good to spend time with different horses too and see the differences in personality, some really enjoy a groom and a fuss, some are much more stand-offish, some are anxious and need really gentle handling, some are cheeky and need a firmer hand, just like children really! As a rider I always take some time before I ride to assess on the ground what sort of mood my horse is in before I ride (particularly an unfamiliar horse), is he feeling tired or energetic that day, is anything worrying him, is he bolshy or submissive to me etc, I find it really helps me know what to look out for and work on when I ride so learning to 'read' a horse in this way will be really helpful to you!